login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A360395 Intersection of A026430 and A360394. 4
1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, 66, 69, 73, 78, 81, 87, 91, 96, 99, 103, 108, 114, 117, 121, 126, 129, 135, 139, 144, 150, 153, 159, 163, 168, 171, 175, 180, 186, 189, 195, 199, 204, 210, 213, 217, 222, 225, 231, 235, 240, 246, 249, 255 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
This is the second of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1) u ^ v = intersection of u and v (in increasing order);
(2) u ^ v';
(3) u' ^ v;
(4) u' ^ v'.
Every positive integer is in exactly one of the four sequences. The limiting densities of these four sequences are 4/9, 2/9, 2/9, and 1/9, respectively.
For A360395, u, v, u', v', are sequences obtained from the Thue-Morse sequence, A026430, as follows:
u = A026530 = (1,3,5,6,8,9,10, 12, ... ) = partial sums of A026430
u' = A356133 = (2,4,7,11,13,17, 20, ... ) = complement of u
v = u + 1 = A285954, except its initial 1
v' = complement of v.
LINKS
EXAMPLE
(1) u ^ v = (3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, ...) = A360394
(2) u ^ v' = (1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, ...) = A360395
(3) u' ^ v = (7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, ...) = A360396
(4) u' ^ v' = (2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, ...) = A360397
MATHEMATICA
z = 400;
u = Accumulate[1 + ThueMorse /@ Range[0, z]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u]; (* A356133 *)
v = u + 2 ; (* A360392 *)
v1 = Complement[Range[Max[v]], v]; (* A360393 *)
Intersection[u, v] (* A360394 *)
Intersection[u, v1] (* A360395 *)
Intersection[u1, v] (* A360396 *)
Intersection[u1, v1] (* A360397 *)
CROSSREFS
Sequence in context: A020147 A316031 A316032 * A316033 A203468 A316034
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 05 2023
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 14 20:49 EDT 2024. Contains 374323 sequences. (Running on oeis4.)